
auseinander
apart
Expresses a spatial separation or distance between two or more things or people.
Die beiden Häuser stehen weit auseinander.
The two houses are far apart.
apart
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apart
Expresses a spatial separation or distance between two or more things or people.
Die beiden Häuser stehen weit auseinander.
The two houses are far apart.

to tell apart
To be able to distinguish two or more similar things or people from one another.
Ich kann die beiden Zwillingsbrüder kaum auseinander halten.
I can hardly tell the two twin brothers apart.

to be apart
Describes the state of people having separated or being in disagreement because of a conflict.
Nach dem großen Streit waren die beiden Freunde eine Weile auseinander.
After the big argument, the two friends were apart for a while.
The word 'auseinander' (apart) is made up of 'aus' (out of) and 'einander' (one another). It literally means 'from one another' or 'out of the group' and always expresses some form of separation or distance.
Nach dem Streit gingen die Freunde im Zorn auseinander.
After the argument, the friends angrily went their separate ways.
'Auseinander-' is a common separable prefix for verbs such as 'auseinandernehmen' (to take apart) or 'auseinandergehen' (to go apart / to separate). In a main clause, the prefix stands at the end.
Der Uhrmacher nimmt das alte Uhrwerk vorsichtig auseinander.
The watchmaker carefully takes the old clockwork apart.
The fixed expression 'jemanden/etwas auseinander halten' (to tell someone/something apart) means to be able to distinguish similar people or things. It is very often used in the negative.
Die eineiigen Zwillinge kann man nur schwer auseinander halten.
It is very hard to tell the identical twins apart.
The phrase 'auseinander sein' (to be apart / to have separated) is a common, rather colloquial way to say that a couple has separated or that friends have fallen out.
Ich habe gehört, dass Maria und Paul schon seit Monaten auseinander sind.
I heard that Maria and Paul have been apart for months.
separate / separated
Auseinander and getrennt can both express spatial separation. Auseinander tends to emphasize movement or the distance between things, whereas getrennt describes the resulting state or a separate arrangement. Use auseinander for a change, and getrennt for an existing state.

Die Kinder gingen nach dem Streit auseinander und setzten sich an verschiedene Tische.
After the argument, the children went their separate ways and sat down at different tables.
Nach dem Streit saßen die Kinder getrennt an verschiedenen Tischen.
After the argument, the children sat separately at different tables.